Proof. By Hölder’s Inequality, for each $y \in [l^{1}(I); X_{i}^{*}]$, $\norm{\phi_y}_{[c_0(I); X_i]^*}\le \norm{y}_{[l^1(I); X_i^*]}$.

Let $\phi \in [c_{0}(I); X_{i}]^{*}$, then there exists $y \in [l^{\infty}(I); X_{i}^{*}]$ such that for each $i \in I$ and $x_{i} \in X_{i}$, $\dpn{x_i \cdot \one_{\bracs{i}}, \phi}{[c_0(I); X_i]}= \dpn{x_i, y_i}{X_i}$.

Let $J \subset I$ be finite and $\alpha \in (0, 1)$, then there exists $x \in [c_{0}(I); X_{i}]$ such that

1. (1)

   $\{i \in I|x_{i} \ne 0\} \subset J$.

2. (2)

   $\norm{x}_{[c_0(I); X_i]}\le 1$.

3. (3)

   For each $j \in J$, $\dpn{x_j, y_j}{X_j}\ge \alpha\norm{y_j}_{X_j^*}$.

Thus

As the above holds for all $\alpha \in (0, 1)$ and $J \subset I$ finite, $y \in [l^{1}(I); X_{i}^{*}]$ with $\norm{y}_{[l^1(I); X_i^*]}\le \norm{\phi}_{[c_0(I); X_i]^*}$. By the Dominated Convergence Theorem, $\dpn{x, \phi_y}{[c_0(I); X_i]}= \dpn{x, \phi}{[c_0(I); X_i]}$ for all $x \in [c_{0}(I); X_{i}]$. Hence the map is an isometric isomorphism.$\square$